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I can’t seem to find the tables that contain median household income for zip codes in American FactFInder.
One reason you may be having trouble finding the tables, is that you may be looking at the American Community Survey rather than the 2010 Census. Zip code data are not yet available via the American Community Survey.
Census tract data is. You can use a crosswalk utility from Mable GeoCORR that will help assign zip code characteristics.
Here’s a tutorial that explains how you can generate zip code data from census tracts.